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\author{王立庆（2022级数学与应用数学1班）}
\title{应用数学前沿专题：教学大纲（学生使用）}
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%\date{2020 年 8 月 28 日}

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\section{时间地点}
\begin{itemize}
\item 上课时间地点：第17-18周，周二上午1-4节、周四上午1-4节，实验中心704. 
\item 答疑时间地点：周二下午5-8节， 一教210. 


\end{itemize}

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\section{教学计划}
\begin{table}[ht]\centering
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\begin{tabular}{|M{1.5cm}|M{0.8cm}|M{4cm}|M{7cm}|}   \hline 
日期		&周	&	参考文献					&	内容		 \\  \hline
6月25日	&17	&	\cite{jms-jh, jms, sauer, sauer-en}	&	Python 简明教程、Numpy 模块		 \\  \hline
6月27日	&17	&	\cite{jms-jh, jms} 		&	二维图形、三维图形、Matplotlib 模块	 \\  \hline
7月2日	&18	&	\cite{jms-jh, jms, sauer, sauer-en} 		&	Scipy 模块、微分方程数值解		 \\  \hline
7月4日	&18	&	\cite{sauer, sauer-en, wallace, nips1989, imageNet, attention, neural-ode, stiff-neural-ode, train-neural-ode} 	&	傅立叶变换与图像压缩、神经网络模型	 	\\  \hline
\end{tabular}
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\section{课程成绩}

    \begin{itemize}
    \item 课堂考勤4次，共30分。
    \item 课堂练习3次，共30分。 
    \item 课外作业1次，共10分。
    \item 期末考试1次，共30分。    
    \end{itemize}


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\section{教学内容}

%\begin{enumerate}

\subsection{Python 简明教程与 Numpy 模块}
\begin{itemize}
\item  {第3章：Python 简明教程。}
理解 Python 的对象、标识符、数值类型、名称空间和类的概念。学会使用列表、元组、字符串、字典这四种容器对象。学会使用判断语句和循环语句。理解函数的语法和不同类型的参数。编写‘分数’这个类。编写程序实现素数的一个筛法。
\item  {第4章：Numpy 模块。}
学会使用 Numpy 的一维数组和二维数组。学会内部和外部的输入输出的方法。学会一些通用的函数和多项式的计算。学会有关线性代数的一些计算。矩阵LU分解，矩阵Cholesky分解，矩阵QR分解，矩阵SVD分解。
\end{itemize}

\subsection{二维图形与三维图形}
\begin{itemize}
\item  {第5章：二维图形。}
理解Matplotlib 的面向对象的编程方式。 学会 Matplotlib 的基本绘图函数，设置曲线样式、标记样式、坐标轴、标签和标题。学会显示文本与数学公式。学会绘制等高线图和复合图形。Mandelbrot 集的可视化。
\item  {第6章：多维图形。}
了解数据降维的方法。了解一些可视化软件。学会孤立波的交互式作图与动画作图。学会三维曲线与曲面的作图。Julia 集的三维可视化。
\item  {第7章：SymPy 模块。}
了解一些计算机代数系统。了解Sympy模块的基本知识。学会矩阵和向量的计算，初等微积分的计算，符号表达式的化简，线性方程组符号求解，常微分方程符号求解。学会用 Sympy 作图。
\end{itemize}

\subsection{微分方程数值解}
\begin{itemize}
\item  {第8章：常微分方程。}
学会使用Python 求解一些常微分方程的初值问题和边值问题。了解延迟微分方程的模型与数值求解。Van de Pol 方程，Sturm-Liouville 问题，Bratu 问题，延迟微分方程。
\item  {第9章：偏微分方程：伪谱方法。}
以伯格斯方程为例，理解偏微分方程的初边值问题，掌握数值解的有限差分方法和谱方法。
\end{itemize}

\subsection{傅立叶变换与图像压缩}
\begin{itemize}
\item 傅立叶级数展开、一维傅立叶变换、一维离散余弦变换DCT.
\item 根据二维DCT来存储图像的例子。
\end{itemize}

\subsection{神经网络模型介绍}
\begin{itemize}
\item 1989年：手写数字识别，卷积神经网络。Yann Lecun. MNIST database. 
\item 2012年：图像识别。Microsoft. Azure open datasets.
\item 2017年：Transformer模型。
\item 2017年：神经网络模型求解常微分方程：PINN模型。
\item 2022年：ChatGPT. 
\end{itemize}



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%\section{数据集}
%\begin{enumerate}
%
%\item Microsoft. Azure open datasets. \\ 
%\url{https://learn.microsoft.com/zh-cn/azure/open-datasets/}
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%\item Yann Lecun. MNIST database. 
%\url{http://yann.lecun.com/exdb/mnist/}
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%\end{enumerate}
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%\section{教材与参考文献}
%    \begin{enumerate}
\begin{thebibliography}{99}
\bibitem{jms-jh} 约翰.M.斯图尔特(著). 江红.余青松(译). \emph{Python科学计算}. 机械工业出版社. 2019年8月第1版. 

\bibitem{jms} John M. Stewart. \emph{Python for Scientists}. Second Edition. Cambridge University Press. 2017. 

\bibitem{sauer} Timothy Sauer(著).裴玉茹.马赓宇(译). \emph{数值分析}. 机械工业出版社. 2018年8月第1版. 

\bibitem{sauer-en} Timothy Sauer. \emph{Numerical Analysis}. Third Edition. Pearson. October 2017. 

\bibitem{wallace}  Gregory K. Wallace.  
\emph{The JPEG Still Picture Compression Standard}. IEEE Transactions on Consumer Electronics.1991. 

\bibitem{nips1989}
LeCun, Yann and Boser, Bernhard and Denker, John and Henderson, Donnie and Howard, R. and Hubbard, Wayne and Jackel, Lawrence. \emph{Handwritten Digit Recognition with a Back-Propagation Network},
Advances in Neural Information Processing Systems, Volume 2,1989. 

\bibitem{imageNet} Alex Krizhevsky, Ilya Sutskever, Geoffrey E. Hinton.  
\emph{ImageNet classification with deep convolutional neural networks}. 
Communications of the ACM, Volume 60 Issue 6, June 2017, pp 84-90. 
%https://doi.org/10.1145/3065386

\bibitem{attention}  Ashish Vaswan, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N. Gomez, Łukasz Kaiser, Illia Polosukhin. 
\emph{Attention is all you need}. arxiv 1706.03762. December 2017. 

\bibitem{neural-ode}
Alexander Norcliffe, Cristian Bodnar, Ben Day, Jacob Moss, and Pietro Lio, Neural ODE processes, 
arxiv:2103.12413v2, August 2021.

\bibitem{stiff-neural-ode}
Suyong Kim, Weiqi Ji, Sili Deng, Yingbo Ma, and Christopher Rackauckas, 
Stiff Neural Ordinary Differential Equations, 
arxiv:2103.15341v3, September 2021. 

\bibitem{train-neural-ode}
Chris Finlay, Jorn-Henrik Jacobsen, Levon Nurbekyan, Adam M Oberman, 
How to Train Your Neural ODE: the World of Jacobian and Kinetic Regularization,
arxiv: 2002.02798v3, June 2020. 

\bibitem{PINNs}
Raissi, M., Perdikaris, P., \& Karniadakis, G. E. (2017). Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378, 686-707.

\end{thebibliography}
%    \end{enumerate}
 
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